Volume-Distance-Ratio Asymptote and Spacetime Inextendibility for FLRW Spacetimes

This paper examines the volume-distance-ratio (VDR) asymptote at the past timelike boundary for Friedman-Lema\^itre-Robertson-Walker (FLRW) spacetimes. We consider spatially flat FLRW spacetimes with scale factor $a(t) \sim t^{\alpha}$, as well as spatially hyperbolic and spherical FLRW spacetimes with scale factor $a(t) \sim a_0 t^{\alpha}$. Using criteria for spacetime inextendibility based on the VDR asymptote, we investigate the conditions under which these FLRW spacetimes are past inextendible.